Linearization of holomorphic vector fields and a characterization of cone singularities
type: Item
instance of: scholarly article
title: Linearization of holomorphic vector fields and a characterization of cone singularities
zbMATH Open document ID: 0641.32020
DOI: 10.1007/BF02941369
author: Wolfgang Richthofer
author: Karl Oeljeklaus
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1988-01-01T00:00:00Z
review text: A local homogeneous cone is a triple (X,0,\(g)\) consisting of a germ of a complex space (X,0) and a finite dimensional Lie algebra \(g\) of holomorphic vector fields on (X,0) satisfying the following conditions: i) \(X\setminus \{0\}\) is smooth ii) \(\zeta (0)=0\) for all \(\zeta\) \(\in g\) and iii) the complex tangent bundle of \(X\setminus \{0\}\) is generated by \(g.\) The main result of the paper is the characterization of the germs (X,0) and the Lie algebras \(g\) with the property that (X,0,\(g)\) is a local homogeneous cone. A local homogeneous cone is shown to be linearizable, i.e. it is equivariantly isomorphic to the germ of the vertex of a homogeneous cone over a projective rational manifold. For the case where (X,0) is smooth, this means that the Lie algebra \(g\) consists of linear vectorfields and homogeneous vectorfields of a fixed degree. The main tool of the proof is the \(g\)-anticanonical map. In particular a special case of the Lipman-Zariski problem in dimension two is settled.
Mathematics Subject Classification ID: 32L05
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 32Sxx
Mathematics Subject Classification ID: 17B15
zbMATH DE Number: 4044209
zbMATH Keywords: linearization of holomorphic vector fields
zbMATH Keywords: cone singularity
zbMATH Keywords: almost homogeneous manifold
reviewed by: Karl Oeljeklaus
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4044209